Wave-particle duality is traditionally viewed as a binary complementarity between interference visibility V and path distinguishability D. In this work, we develop a unified continuous framework where "wave-like" and "particle-like" behaviors are represented as basis-dependent features of a single quantum state, smoothly interpolated by the fractional Fourier transform (FrFT) and broader metaplectic groups. We introduce two operational indices: a coherence index 𝐶(𝛼), defined as the ℓ₁-norm of off-diagonal density matrix elements in the 𝛼-representation, and a localization index 𝐿(𝛼), derived from the Shannon entropy or Fisher information of the 𝛼-domain distribution. We prove that for Gaussian superpositions, these indices reveal a continuous trade-off that generalizes the Englert-Greenberger-Yasin relation. Furthermore, we establish a state-independent entropic uncertainty bound for linear canonical transforms (LCTs) and propose experimental protocols using photonic and matter-wave platforms to map trajectories in the (C, L) plane.
Ying Ye (2026) studied this question.